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finite groups
Paul Lescot, Hung Ngoc Nguyen, Yong Yang
To cite this version:
Paul Lescot, Hung Ngoc Nguyen, Yong Yang. On the commuting probability and supersolvability of finite groups. 2013. �hal-00878343v2�
ON THE COMMUTING PROBABILITY AND SUPERSOLVABILITY OF FINITE GROUPS
PAUL LESCOT (CORRESPONDING AUTHOR) LMRS,CNRS UMR 6085
UFR DES SCIENCES ET TECHNIQUES, UNIVERSIT´E DE ROUEN AVENUE DE L’UNIVERSIT´E BP12
76801 SAINT-ETIENNE DU ROUVRAY, FRANCE PHONE 00 33 (0)2 32 95 52 24
FAX 00 33 (0)2 32 95 52 86
E-MAIL PAUL.LESCOT@UNIV-ROUEN.FR
HUNG NGOC NGUYEN DEPARTMENT OF MATHEMATICS
THE UNIVERSITY OF AKRON AKRON, OHIO 44325, USA
E-MAIL HUNGNGUYEN@UAKRON.EDU
YONG YANG
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN–PARKSIDE
KENOSHA, WI 53141, USA E-MAIL YANGY@UWP.EDU
Date: October 31, 2013.
2010Mathematics Subject Classification. Primary 20E45; Secondary 20D10.
Key words and phrases. finite group, conjugacy class, commuting probability.
1
Abstract. For a finite groupG, letd(G) denote the probability that a randomly chosen pair of elements of G commute. We prove that if d(G) > 1/s for some integers > 1 andG splits over an abelian normal nontrivial subgroupN, thenG has a nontrivial conjugacy class insideN of size at mosts−1. We also extend two results of Barry, MacHale, and N´ı Sh´e on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that ifd(G)>5/16 then eitherGis supersolvable, orGisoclinic toA4, orG/Z(G) is isoclinic toA4.
1. Introduction
For a group G, let d(G) denote the probability that a randomly chosen pair of elements of Gcommute. That is,
d(G) := 1
|G|2|{(x, y)∈G×G|xy =yx}|.
This quantity is often referred to as the commuting probability of G. The study of the commuting probability of finite groups dates back to work of Gustafson in the seventies. In [6], he showed that
d(G) = k(G)
|G| , where k(G) is the number of conjugacy classes ofG.
It is clear thatd(G) = 1 if and only if G is abelian. Therefore, whend(G) is close to 1, one might expect that G is close to abelian. For instance, it was proved by Gustafson in the same paper that if d(G) > 5/8 then G must be abelian. In [12], the first author classified all groups with commuting probability at least 1/2 – if d(G) ≥ 1/2 then G is isoclinic to the trivial group, an extraspecial 2-group, or S3. As a consequence, if d(G) > 1/2 then G must be nilpotent. Going further, the first author proved in [10, 11] that G is solvable whenever d(G) > 1/12. This was improved by Guralnick and Robinson in [5, Theorem 11] where they showed that if d(G) > 3/40 then either G is solvable or G ∼= A5 ×A for some abelian group A.
In [1], Barry, MacHale, and N´ı Sh´e proved that G must be supersolvable whenever d(G)>1/3 and pointed out that, sinced(A4) = 1/3, the bound cannot be improved.
Two finite groups are said to beisoclinic if there exists isomorphisms between their inner automorphism groups and commutator subgroups such that these isomorphisms are compatible with the commutator map, see §3 for the detailed definition. This concept is weaker than isomorphism and was introduced by Hall [7] in connection with the enumeration ofp-groups. It was shown by the first author [12] that the commuting probability is invariant under isoclinism. It follows that any group isoclinic toA4 has commuting probability exactly equal to 1/3. Our first result highlights the special role ofA4 among non-supersolvable groups with commuting probability greater than 5/16. Here and what follows, the center of G, as usual, is denoted byZ(G).
THE COMMUTING PROBABILITY OF FINITE GROUPS 3
Theorem 1. Let G be a finite group. If d(G)>5/16, then (i) G is supersolvable, or
(ii) G is isoclinic to A4, or (iii) G/Z(G) is isoclinic to A4.
Theorem 1 has two consequences. The obvious one is the aforementioned result of Barry, MacHale, and N´ı Sh´e. We would like to note that their proof is somewhat more complicated and requires a large amount of computations with GAP [4]. The second one is less obvious and shows that the groups isoclinic to A4 are the only non-supersolvable groups of commuting probability at least 1/3.
Corollary 2. Let Gbe a finite group withd(G)≥1/3, thenG is either supersolvable or isoclinic to A4.
We remark that, the average size of a conjugacy class of G, denoted by acs(G), is exactly the reciprocal of d(G). Therefore, Theorem 1 is equivalent to: that if acs(G) < 16/5 then either G is supersolvable or G is isoclinic to A4 or G/Z(G) is isoclinic to A4. Recently, Isaacs, Loukaki, and Moret´o [9] have obtained some dual results on solvability and nilpotency in connection with average character degree of finite groups. For instance, they showed that a finite group is supersolvable whenever its average character degree is less than 3/2. It would be interesting if there is a dual result of Theorem 1 for the average character degree.
For groups of odd order, it is possible to obtain better bounds. It was proved in [1]
that if G is a group of odd order with d(G)>11/75, then Gmust be supersolvable.
Let Cn denote the cyclic group of order n. We notice that (C5 ×C5)⋊C3 is the smallest non-supersolvable group of odd order. Here we can show that the groups isoclinic to (C5 ×C5)⋊C3 have commuting probability ‘substantially’ larger than that of other non-supersolvable groups of odd order.
Theorem 3. Let G be a finite group of odd order. If d(G)>35/243 <11/75, then G is either supersolvable or isoclinic to (C5×C5)⋊C3.
Our last result provides a characteristic of certain groups with ‘large’ commuting probability and therefore can be applied to obtain the inside structure of these groups.
For an example, see §4.
Theorem 4. Let s ≥ 2 be an integer and G a finite group with d(G) > 1/s. Let N be an abelian normal nontrivial subgroup of G and suppose that G splits over N. Then there exists a nontrivial conjugacy class of G inside N of size at most s−1.
In particular, we have either Z(G)6= 1 or G has a proper subgroup of index at most s−1.
Theorems 1, 3 and 4 are respectively proved in Sections 2, 3, and 4. Corollary 2 is proved at the end of Section 2.
2. Groups with commuting probability greater than 5/16
We will prove Theorem 1 and Corollary 2 in this section. We first recall a well- known result of Gallagher [3].
Lemma 5. If N is a normal subgroup of G, then k(G)≤k(G/N)k(N) and the equality is equivalent to
CG/N(gN) =CG(g)N/N for each g ∈G.
This gives an immediate consequence.
Lemma 6. Let N be a normal subgroup of G. Then (i) d(G)≤d(G/N)d(N),
(ii) d(G) = d(G/N) if and only if N is abelian and CG/N(gN) = CG(g)N/N for each g ∈G, and
(iii) if N ⊆Z(G) and d(G) =d(G/N), then Z(G/N) =Z(G)/N.
Proof. (i) and (ii) are consequences of Lemma 5. We now prove (iii). Assume that N ⊆Z(G) and d(G) = d(G/N). We have CG/N(gN) = CG(g)N/N for every g ∈ G and therefore
gN ∈Z(G/N)⇔CG/N(gN) =G/N
⇔CG(g)N =G
⇔CG(g) =G
⇔g ∈Z(G)
Therefore, Z(G/N) =Z(G)/N, as desired.
Two groups G and H are said to be isoclinic if there are isomorphisms ϕ : G/Z(G)→H/Z(H) and φ:G′ →H′ such that
if ϕ(g1Z(G)) = h1Z(H) and ϕ(g2Z(G)) = h2Z(H),
then φ([g1, g2]) = [h1, h2].
This concept is weaker than isomorphism and was introduced by Hall in [7] as a structurally motivated classification for finite groups, particularly for p-groups. It is well-known that several characteristics of finite groups are invariant under isoclinism and in particular supersolvability is one of those, see [2]. Furthermore, it is proved in [12] that the commuting probability is also invariant under isoclinism.
A stem group is defined as a group whose center is contained inside its derived subgroup. It is known that every group is isoclinic to a stem group and if we restrict to finite groups, a stem group has the minimum order among all groups isoclinic to
THE COMMUTING PROBABILITY OF FINITE GROUPS 5
it, see [7] for more details. The following lemma plays an important role in the proof of Theorems 1 and 3.
Lemma 7. For every finite group G, there is a finite group H isoclinic to G such that |H| ≤ |G| and Z(H)⊆H′.
The next lemma will narrow down the possibilities for the commutator subgroup of a finite group with commuting probability greater than 5/16.
Lemma 8. Let G be an finite group with d(G)>5/16. Then |G′|<12.
Proof. Let Irr2(G) denote the set of nonlinear irreducible complex characters of G.
Then, asGhas exactly [G:G′] linear characters, we have|Irr2(G)|=k(G)−[G:G′] where k(G) is the number of conjugacy classes ofG. We obtain
|G|= [G:G′] + X
χ∈Irr2(G)
χ(1)2 ≥[G:G′] + 4(k(G)−[G:G′]).
As d(G) =k(G)/|G|, it follows that 1
|G′|+ 4(d(G)− 1
|G′|)≤1.
Using the hypothesis d(G)>5/16, we deduce that |G′|<12.
We are now ready to prove Theorem 1.
Proof of Theorem 1. Assume that Gis a finite group with d(G)>5/16 and Gis not supersolvable. We aim to show that either G is not isoclinic to A4 or G/Z(G) is isoclinic toA4. Since commuting probability and supersolvability are both invariant under isoclinism, using Lemma 7, we can assume that Z(G)⊆G′. Indeed, if Z(G) = G′, then G is nilpotent which violates our assumption. So we assume furthermore that Z(G) G′. Recall that d(G) > 5/16 and hence |G′| ≤ 11 by Lemma 8. We note that G′ is noncyclic as Gis not supersolvable.
First we remark thatS3, D8 as well asD10 have a cyclic, characteristic, non-central subgroup and hence it is well-known that they cannot arise as commutator subgroups, see [13] for instance. Thus we are left with the following possibilities of G′.
Case G′ ∼= C2 ×C2: If Z(G) ∼= C2 then the normal series 1 < Z(G) < G′ < G implies thatG is supersolvable. So we assume that Z(G) = 1. Thus G′ is a minimal normal subgroup of G. Now, since G is not supersolvable, G′ * Φ(G), see [8] for instance. Therefore,G′ is not contained in a maximal subgroup G, say M. We have G =G′M. Also, as G′ is abelian, we see that G′∩M G. Now the minimality of G′ and the fact that G′ is not contained in M imply that G′ ∩M = 1. This means G splits over G′ or equivalently G ∼= G′ ⋊M. Thus, as M ∼= G/G′ is abelian and Z(G) = 1, we deduce thatCM(G′) = 1. It follows that
M ≤Aut(G′)∼= Aut(C2×C2)∼=S3,
and hence
M ∼=C2 or M ∼=C3.
In the former case, |G|= 8 andG would be nilpotent, a contradiction. In the latter case, G∼=A4 and we are done.
Case G′ ∼= C3 ×C3: As in the previous case, we can assume Z(G) = 1, G′ is a minimal normal subgroup of G, and G ∼= G′ ⋊ M. Here M is an abelian subgroup of Aut(G′) ∼= GL2(3). Consulting the list of subgroups of GL2(3) reveals:
M ∼= C2, C3, C4, C2 ×C2, C6, or C8. However, it is routine to check that all these possibilities result in eitherG is supersolvable or d(G)≤5/16.
In the remaining cases, we letN be a minimal normal subgroup ofGwithN ⊆G′. Recall that Z(G) G′ and so in the caseZ(G)6= 1, we can even take N ⊆Z(G).
Case G′ ∼= Q8, C4 ×C2, or C2 ×C2 ×C2 and N ∼= C2: If N < Z(G) then the normal series 1 < N < Z(G) < G′ < G would imply that that G is supersolvable, a contradiction. Thus Z(G) = N ∼=C2. Also, as G is not supersolvable, G/Z(G) is not supersolvable as well. It follows that (G/Z(G))′ is not cyclic so that
( G
Z(G))′ ∼=C2×C2.
By Lemma 6, we know thatd(G/N)≥d(G)>5/16. Now we are in the first case with G/Z(G) replacing G. Therefore, we conclude that either G/Z(G) is supersolvable, a contradiction, or G/Z(G) is isoclinic toA4, as desired.
Case G′ ∼= C4×C2, or C2 ×C2×C2 and N ∼= C2×C2: Then G′/N ∼= C2 is a normal subgroup of G/N. In particular, G′/N ⊆ Z(G/N) and as G/G′ is abelian, we deduce that G/N is supersolvable. As in the previous case, by using the non- supersolvability of G, we deduce that G∼= N ⋊M where M is a maximal subgroup of G. It then follows that C2 ∼= G′ ∩M ⊳ M and hence G′ ∩M centralizes M, whenceG′∩M centralizes G. This implies thatZ(G)6= 1, which in turn implies that N =Z(G) since N ⊆Z(G)(G′. This violates the minimality of N.
Case N =G′ ∼=C2×C2×C2: SinceZ(G) G′, we obtain Z(G) = 1. As before, we can show thatG∼=G′⋊M whereM is an abelian subgroup of Aut(G′)∼= GL3(2).
This implies thatM ∼=C2, C3, C4, C2×C2, or C7. The cases M ∼=C2, C4 or C2×C2
would imply that G is a 2-group. The case M ∼= C7 would imply that d(G) = 1/7.
Finally, the case M ∼=C3 implies thatG∼=C2×A4 and hence G′ ∼=C2 ×C2, which
is the final contradiction.
Now we prove Corollary 2.
Proof of Corollary 2. Assume, to the contrary, that the statement is false and let G be a minimal counterexample. Again we know that Z(G) G′ and as in the proof of Lemma 8, we also have |G′| ≤ 9. Using Theorem 1, we deduce that G/Z(G) is
THE COMMUTING PROBABILITY OF FINITE GROUPS 7
isoclinic to A4 and Z(G) is nontrivial. In particular, G′/Z(G) ∼=A′4 ∼=C2×C2 and hence Z(G)∼=C2. Using Lemma 6(i), we have
1
3 =d( G
Z(G))≥d(G)≥ 1 3.
Therefore d(G) =d(G/Z(G)), which implies that Z(G/Z(G)) =Z(G)/Z(G) = 1 by Lemma 6(iii). It follows thatG/Z(G)∼=A4 and as|Z(G)|= 2, we haveG=C2×A4.
This violates the assumption thatZ(G) G′.
3. Groups of odd order
We will prove Theorem 3 in this section. As in Section 2, we narrow down the possibility for the order of the commutator subgroup of a group in consideration.
Lemma 9. Let G be an odd order finite group with d(G)>35/243. Then |G′|<27.
Proof. We repeat some of the arguments in the proof of Lemma 8. Recall that Irr2(G) denotes the set of nonlinear irreducible complex characters ofGand we have
|Irr2(G)| =k(G)−[G : G′]. Since |G| is odd, every character in Irr2(G) has degree at least 3. We obtain
|G|= [G:G′] + X
χ∈Irr2(G)
χ(1)2 ≥[G:G′] + 9(k(G)−[G:G′]),
and therefore
1
|G′|+ 9(d(G)− 1
|G′|)≤1.
Since d(G)>35/243, it follows that |G′|<27, as wanted.
Proof of Theorem 3. We argue by contradiction and let G be a minimal counterex- ample. Since commuting probability and supersolvability are both invariant under isoclinism, using Lemma 7, we can assume that Z(G) ⊆ G′. Now G is a non- supersolvable group of odd order with d(G)>35/243. Applying Lemma 9, we have
|G′|<27 so that G′ is a noncyclic odd order group of order at most 25.
We choose a minimal normal subgroup N of G with N ⊆ G′. and note that N is elementary abelian. By Lemma 6, we have d(G/N) ≥ d(G) > 35/243 so that G/N is supersolvable or isoclinic to (C5×C5)⋊C3 by the minimality of G.
First we show that the case whereG/N is isoclinic to (C5×C5)⋊C3cannot happen.
Assume so. Then
G′/N = (G/N)′ ∼= ((C5×C5)⋊C3)′ =C5×C5,
which implies that |G′| is at least 50, a contradiction. We conclude that G/N is supersolvable. In particular, if N is cyclic then G is supersolvable and we have a contradiction. Note that, when G′ ≇C3×C3 and C5×C5, a routine check on groups of odd order at most 25 shows that N must be cyclic. Thus, it remains to consider
the cases where G′ ∼=C3×C3 orG′ ∼=C5×C5 and G′ is a minimal normal subgroup of G.
IfG′ ⊆Φ(G) then G is nilpotent and we are done. Therefore we can assume that G′ * Φ(G) and, as in the proof of Theorem 1, we see that G splits over G′ by a maximal subgroup M of G:
G∼=G′⋊M.
Since Z(G)⊆ G′, we must have Z(G) = 1 and, as M ∼= G/G′ is abelian, we deduce that CM(G′) = 1.It follows that
M ≤Aut(G′).
First we assume thatG′ ∼=C3×C3, then M is an abelian subgroup of odd order of Aut(C3 ×C3) = GL2(3). This forces M ∼= C3, which implies that |G| = |G′||M| = 9·3 = 27 and henceGis nilpotent, a contradiction. Next we assume thatG′ ∼=C5×C5. Arguing similarly, we see thatM is an abelian subgroup of odd order of GL2(5). This forcesM ∼=C3, C5 orC15. The caseM ∼=C5 would imply that Gis nilpotent whereas the case M ∼= C15 would imply that d(G) = 23/375, which is a contradiction. We conclude that M ∼=C3 so that G∼= (C5×C5)⋊C3.
4. A conjugacy class size theorem
In this section, we prove Theorem 4 and then give an example showing how one can obtain some properties of certain groups with ‘large’ commuting probability.
Proof of Theorem 4. Assume, to the contrary, that all the nontrivial orbits of the conjugacy action of G onN have size at least s. Since Gsplits over N, let G=HN where H∩N = 1. We denote C =CH(N) and clearlyC H. Every element of G can be written uniquely as ha where h ∈ H and a ∈ N. We now examine the class sizes in G.
First, letg =ha witha6= 1. Sinea is in an orbit ofH onN of size greater than or equal to 3, we can find s−1 other elements a2, a3, ..., as in the orbit of a. Therefore there exist t2, t3, ..., ts∈H such that ati =ai for 2≤i≤s. Thus
g =ha, gt2 =ht2a2, gt3 =ht3a3, ..., gts =htsas
are s different elements in the conjugacy class of g. We now have
(4.1) every conjugacy class of an element outside H has size at least s.
It remains to consider the conjugacy classes of elements of H. Letg =h1 for some h1 ∈H. If h1 6∈C, then there exits some a ∈N which is not fixed by h1. Thus
ah1a−1 =h1h−11ah1a−1 =h1a1, where a1 =h−11ah1a−1 6= 1.
By the previous paragraph, we know that there aret2, t3, ..., ts∈H such that g =h1, h1a1, ht12a2, ht13a3, ..., ht1sas
THE COMMUTING PROBABILITY OF FINITE GROUPS 9
are s+ 1 different elements in the conjugacy class ofg, where ai’s are nontrivial and distinct. Thus g is in a conjugacy class of size at leasts+ 1.
Ifh1 6=h2,h1, h2 ∈H\C, andh1, h2 are in the same conjugacy class, then we know that
h1, h1a1, ht12a2, ht13a3, ..., ht1sas, h2
are s + 2 distinct elements in the same conjugacy class. This implies in general that if h1, h2, . . . , ht are distinct elements in H\C and h1, h2, . . . , ht are in the same conjugacy class, then the size of the conjugacy class is greater than or equals to
s+t.
Denote k = |H|/|C|, then there are (k −1)|C| elements in H\C. We consider all the conjugacy classes of G which contain some elements in H. Suppose all the elements in H\C belong to n different conjugacy classes and each conjugacy class contains t1, . . . , tn elements in H\C respectively. Then
Xn
i=1
ti =|H\C|= (k−1)|C|
and the sum of sizes of these n classes is at least Xn
i=1
(s+ti) =ns+ Xn
i=1
ti.
Therefore, the average size of a conjugacy class of an element inH is at least ns+Pn
i=1ti+|C|
n+|C| = ns+k|C|
n+|C| .
Since C acts trivially on N and |H/C| = k, the conjugacy action of G on N has orbit of size at most k. Since N is nontrivial and every nontrivial orbit of G on N has size at least s, we deduce that k ≥s. It follows that
ns+k|C|
n+|C| ≥ ns+s|C|
n+|C| =s and hence
(4.2) the average size of a class of an element in H is at least s.
Combining (4.1) and (4.2), we conclude that the average class size ofG is at least s,
which violates the hypothesis that d(G)>1/s.
The following is an application of Theorem 4 to the study of finite groups with commuting probability greater than 1/3.
Corollary 10. Let G = (C2×C2)⋊H and assume that d(G) > 1/3. Then there exists a nontrivial element of C2 ×C2 that is fixed under the conjugation action of H. In particular, Z(G)6= 1.
Proof. By Theorem 4, the groupH has a nontrivial orbit of size at most 2 onC2×C2. If this orbit has size 1 then we are done. Otherwise, it has size 2 and hence the other
orbit must have size 1, as wanted.
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